direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C2×C8.12D4, C42.359D4, C42.713C23, C8.52(C2×D4), (C4×C8)⋊74C22, (C2×C8).261D4, C4.4(C22×D4), (C22×D8).9C2, C4.14(C4⋊1D4), (C2×C4).344C24, (C2×C8).593C23, (C22×Q16)⋊11C2, (C2×Q16)⋊44C22, (C22×C4).566D4, C23.879(C2×D4), (C2×Q8).98C23, (C2×SD16)⋊78C22, (C22×SD16)⋊27C2, (C2×D8).129C22, (C2×D4).110C23, C22.98(C4○D8), C4.4D4⋊56C22, C22.50(C4⋊1D4), (C22×C8).537C22, C22.604(C22×D4), (C22×C4).1559C23, (C2×C42).1128C22, (C22×D4).373C22, (C22×Q8).306C22, (C2×C4×C8)⋊29C2, C2.30(C2×C4○D8), (C2×C4).854(C2×D4), C2.23(C2×C4⋊1D4), (C2×C4.4D4)⋊40C2, SmallGroup(128,1878)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 628 in 296 conjugacy classes, 116 normal (16 characteristic)
C1, C2, C2 [×6], C2 [×4], C4 [×2], C4 [×2], C4 [×8], C22, C22 [×6], C22 [×20], C8 [×8], C2×C4 [×2], C2×C4 [×8], C2×C4 [×12], D4 [×12], Q8 [×12], C23, C23 [×16], C42 [×4], C22⋊C4 [×16], C2×C8 [×12], D8 [×8], SD16 [×16], Q16 [×8], C22×C4, C22×C4 [×2], C22×C4 [×2], C2×D4 [×4], C2×D4 [×10], C2×Q8 [×4], C2×Q8 [×10], C24 [×2], C4×C8 [×4], C2×C42, C2×C22⋊C4 [×4], C4.4D4 [×8], C4.4D4 [×4], C22×C8 [×2], C2×D8 [×4], C2×D8 [×4], C2×SD16 [×8], C2×SD16 [×8], C2×Q16 [×4], C2×Q16 [×4], C22×D4 [×2], C22×Q8 [×2], C2×C4×C8, C8.12D4 [×8], C2×C4.4D4 [×2], C22×D8, C22×SD16 [×2], C22×Q16, C2×C8.12D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C24, C4⋊1D4 [×4], C4○D8 [×4], C22×D4 [×3], C8.12D4 [×4], C2×C4⋊1D4, C2×C4○D8 [×2], C2×C8.12D4
Generators and relations
G = < a,b,c,d | a2=b8=c4=1, d2=b4, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b3, dcd-1=b4c-1 >
►On 64 points
(1 16)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(8 15)(17 55)(18 56)(19 49)(20 50)(21 51)(22 52)(23 53)(24 54)(25 48)(26 41)(27 42)(28 43)(29 44)(30 45)(31 46)(32 47)(33 62)(34 63)(35 64)(36 57)(37 58)(38 59)(39 60)(40 61)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)
(1 32 17 63)(2 25 18 64)(3 26 19 57)(4 27 20 58)(5 28 21 59)(6 29 22 60)(7 30 23 61)(8 31 24 62)(9 48 56 35)(10 41 49 36)(11 42 50 37)(12 43 51 38)(13 44 52 39)(14 45 53 40)(15 46 54 33)(16 47 55 34)
(1 59 5 63)(2 62 6 58)(3 57 7 61)(4 60 8 64)(9 33 13 37)(10 36 14 40)(11 39 15 35)(12 34 16 38)(17 28 21 32)(18 31 22 27)(19 26 23 30)(20 29 24 25)(41 53 45 49)(42 56 46 52)(43 51 47 55)(44 54 48 50)
G:=sub<Sym(64)| (1,16)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(8,15)(17,55)(18,56)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(25,48)(26,41)(27,42)(28,43)(29,44)(30,45)(31,46)(32,47)(33,62)(34,63)(35,64)(36,57)(37,58)(38,59)(39,60)(40,61), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,32,17,63)(2,25,18,64)(3,26,19,57)(4,27,20,58)(5,28,21,59)(6,29,22,60)(7,30,23,61)(8,31,24,62)(9,48,56,35)(10,41,49,36)(11,42,50,37)(12,43,51,38)(13,44,52,39)(14,45,53,40)(15,46,54,33)(16,47,55,34), (1,59,5,63)(2,62,6,58)(3,57,7,61)(4,60,8,64)(9,33,13,37)(10,36,14,40)(11,39,15,35)(12,34,16,38)(17,28,21,32)(18,31,22,27)(19,26,23,30)(20,29,24,25)(41,53,45,49)(42,56,46,52)(43,51,47,55)(44,54,48,50)>;
G:=Group( (1,16)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(8,15)(17,55)(18,56)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(25,48)(26,41)(27,42)(28,43)(29,44)(30,45)(31,46)(32,47)(33,62)(34,63)(35,64)(36,57)(37,58)(38,59)(39,60)(40,61), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,32,17,63)(2,25,18,64)(3,26,19,57)(4,27,20,58)(5,28,21,59)(6,29,22,60)(7,30,23,61)(8,31,24,62)(9,48,56,35)(10,41,49,36)(11,42,50,37)(12,43,51,38)(13,44,52,39)(14,45,53,40)(15,46,54,33)(16,47,55,34), (1,59,5,63)(2,62,6,58)(3,57,7,61)(4,60,8,64)(9,33,13,37)(10,36,14,40)(11,39,15,35)(12,34,16,38)(17,28,21,32)(18,31,22,27)(19,26,23,30)(20,29,24,25)(41,53,45,49)(42,56,46,52)(43,51,47,55)(44,54,48,50) );
G=PermutationGroup([(1,16),(2,9),(3,10),(4,11),(5,12),(6,13),(7,14),(8,15),(17,55),(18,56),(19,49),(20,50),(21,51),(22,52),(23,53),(24,54),(25,48),(26,41),(27,42),(28,43),(29,44),(30,45),(31,46),(32,47),(33,62),(34,63),(35,64),(36,57),(37,58),(38,59),(39,60),(40,61)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64)], [(1,32,17,63),(2,25,18,64),(3,26,19,57),(4,27,20,58),(5,28,21,59),(6,29,22,60),(7,30,23,61),(8,31,24,62),(9,48,56,35),(10,41,49,36),(11,42,50,37),(12,43,51,38),(13,44,52,39),(14,45,53,40),(15,46,54,33),(16,47,55,34)], [(1,59,5,63),(2,62,6,58),(3,57,7,61),(4,60,8,64),(9,33,13,37),(10,36,14,40),(11,39,15,35),(12,34,16,38),(17,28,21,32),(18,31,22,27),(19,26,23,30),(20,29,24,25),(41,53,45,49),(42,56,46,52),(43,51,47,55),(44,54,48,50)])
Matrix representation ►G ⊆ GL5(𝔽17)
| 16 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 10 | 10 |
| 0 | 0 | 0 | 12 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 13 | 13 |
G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,0,16,0,0,0,1,0,0,0,0,0,0,10,12,0,0,0,10,0],[1,0,0,0,0,0,0,16,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,4],[16,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,4,13,0,0,0,0,13] >;
44 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4L | 4M | 4N | 4O | 4P | 8A | ··· | 8P |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 8 | 8 | 8 | 8 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 2 | ··· | 2 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | C4○D8 |
| kernel | C2×C8.12D4 | C2×C4×C8 | C8.12D4 | C2×C4.4D4 | C22×D8 | C22×SD16 | C22×Q16 | C42 | C2×C8 | C22×C4 | C22 |
| # reps | 1 | 1 | 8 | 2 | 1 | 2 | 1 | 2 | 8 | 2 | 16 |
In GAP, Magma, Sage, TeX
C_2\times C_8._{12}D_4 % in TeX
G:=Group("C2xC8.12D4"); // GroupNames label
G:=SmallGroup(128,1878);
// by ID
G=gap.SmallGroup(128,1878);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,520,2804,172]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^8=c^4=1,d^2=b^4,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^3,d*c*d^-1=b^4*c^-1>;
// generators/relations